Error propagation in adaptive Simpson algorithm

[Albert Chan]

Hi, Claudio

The problem with the sin integral is not cancellation, but its periodic nature.
Simpson's Rule is also based on periodic sampling (all equally spaced).

If the sampling and periodic function were in sync, the samples are going to be biased.
For sin(x) 0 to 200, the first few iterations, none of sin(x) samples were above zero.

A non-linear transformed function can fix this, by scrambling the sample points.
Have you tried the non-linear transformed sin integral (same thread, post #13) ?

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Regarding "excessive" accuracy, it will happen even if tolerance is good estimater for accuracy.
Not all integral have the same convergence rate.

Say, tolerance of 1e-8 somehow guaranteed 7 digits accuracy (it does not)
What is the chance of really getting 7 digits accuracy ? Almost zero.

With guaranteed minimum accuracy, average accuracy is going to be higher, say, 10 digits.
Iterations that not quite make it to tolerance will doubled the points, "wasting" accuracy.